[R] glmx specification of heteroskedasticity (and its use in Heckit)
Achim.Zeileis at uibk.ac.at
Sat Jun 1 00:29:40 CEST 2013
On Fri, 31 May 2013, Michal Kvasni?ka wrote:
> Many thanks for your answer. Let me check please that I do understand
> it correctly. Does it mean that the estimated log-likelyhood function
> is (in the Gaussian case)
> sum y * log F(x'b / exp(z'g)) + sum (1 - y) * log(1 - F(x'b / exp(z'g))
> where F is standard normal CDF, and the rest is as in your mail?
Yes, this is the likelihood.
In a GLM context, one would call this the "Gaussian" case though. It's the
binomial case with a probit link: family = binomial(link = "logit"). And
this is equivalent to observing a binary variable from a latent Gaussian.
However, it would also be possible to set family = gaussian where the
likelihood itself would be Gaussian (typically with an identity link).
> Many thanks once more.
> Best wishes
> P.S. Sorry if you get this mail twice -- I'm not yet certain with this
> mailing list to what mail address I should reply.
> 2013/5/31 Achim Zeileis <Achim.Zeileis at uibk.ac.at>:
>> On Fri, 31 May 2013, Michal Kvasni?ka wrote:
>>> First many thanks to its authors for glmx package and hetglm()
>>> function especially. It is absolutely great.
>> Glad it is useful for you!
>>> Now, let me ask my question: what model of heteroskedasticity hetglm()
>>> uses? Is the random part of the Gaussian probit model
>>> norm(0, sd = exp(X2*beta2))
>>> where norm is the Gaussian distribution, 0 is its zero mean, and sd is
>>> its standard deviation modelled as a linear model with explanatory
>>> variables X2 (a matrix) and some unknown parameters beta2?
>> In the hetglm model the response y is distributed with mean mu and from some
>> exponential family (default: binomial). And the following equation holds:
>> mu = h( x'b / exp(z'g) )
>> where h() is the inverse link function. Thus if h() is the normal
>> distribution function (inverse probit link), then
>> mu = P(X > 0)
>> where X is normally distributed with mean x'b and standard deviation
>> Hope that helps,
>>> I'm asking because after estimating a heteroskedastic probit, I want
>>> to estimate a Heckit. I plan to use two-stage estimation procedure. In
>>> the first step I want to estimate the heteroskedastic probit, and in
>>> the second step the linear part (with bootstrapped confidence
>>> intervals of parameters). The linear part includes inverse Mill's
>>> ration lambda where
>>> lambda = dnorm(X1*beta1, sd=?) / pnorm(X1*beta1, sd=?)
>>> where X1 are the explanatory variables of the probit model, and beta1
>>> are their parameters. (I hope I can tweak the homoskedastic model this
>>> way.) (I plan to use two-step estimation to avoid further distribution
>>> assumptions on the linear part of the model.)
>>> Many thanks for your answer to my question (and also for any comment
>>> on the overall estimation procedure).
>>> Best wishes,
>>> R-help at r-project.org mailing list
>>> PLEASE do read the posting guide
>>> and provide commented, minimal, self-contained, reproducible code.
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